My question is about understanding a remark John Conway made in On Numbers and Games (ONAG), where he proposes a method for constructing the real numbers from the rationals. I will have to assume familiarity with Conway's construction of the surreal number field.
The context of this quote is Conway discussing of an unfortunate feature of constructing $\mathbb R$ from Dedekind cuts of $\mathbb Q$. Namely, when defining multiplication $x\cdot y$ of real numbers, you need to break into four cases based on the signs of $x$ and $y$. This means that proving associativity requires breaking into eight cases, which is clunky. One could instead modify Conway's construction of the surreals to construct $\mathbb R$, adding an extra axiom so the extra infinities and infinitesimals are not constructed. This solves the problem clunky associativity proof, since multiplication in the surreal numbers is defined without need for case splitting. However, the full surreal number construction has some drawbacks (being difficult to understand for undergraduates, and giving special status to the dyadic rationals).
To this end, Conway proposes the following compromise:
There is another way out. If we adopt a classical approach as far as the construction of $\mathbb Q$, and the define the reals as Dedekind sections of $\mathbb Q$ with the definitions of addition and multiplication given in this book, then all formal laws have 1-line proofs and there is no case splitting.
I surmise this means that real numbers would be defined as ordered pairs $(L,R)$ of sets of rational numbers, where for all $\ell \in L$ and $r\in R$ we have $\ell < r$, and $L\cup R$ excludes at most one member of $\mathbb Q$. My question is, how does Conway's suggestion actually work, in detail?
That is the end of my question, what follows are my thoughts on the problem.
Following the surreal definition of multiplication, letting $x$ and $y$ be real numbers, and letting $x^L$ and $x^R$ be variables which range over the left and right sections defining $x$, it seems that Conway would define $$ xy=(\{x^Ly+xy^L-x^Ly^L,x^Ry+xy^R-x^Ry^R\},\{x^Ly+xy^R-x^Ly^R,x^Ry+xy^L-x^Ry^L\})\tag1 $$ However, this raises the question, "what does $x^Ly$ mean?". This is the product of a rational with a real number. You cannot use the above definition to define this product, since Conway stipulated the rationals were to be constructed as normal, so they do not have a left and right section. However, it seems the only sensible way to define the multiplication of a rational with $q$ with a real number $x$ would involve case-splitting, as below, defeating the whole purpose. $$ xq= \begin{cases} (\{x^Lq\},\{x^Rq\}) & q>0\\ (\{x^Rq\},\{x^Lq\}) & q<0\\ 0 & q=0 \end{cases} $$
The same problem would occur for the definition of addition: $$ x+y=(\{x+y^L,x^L+y\},\{x+y^R,x^R+y\}) $$ However, there is a nice workaround, where you instead define $$ x+y=(\{x^L+y^L\},\{x^R+y^R\}) $$ This is equivalent to the usual definition of addition in Dedekind's construction of the reals. It works because $x^L<x$ and $y^L<y$ implies $x^L+y^L<x+y$, so that $x^L+y^L$ can be safely put in the left section of $x+y$. Is there a similar fix for the multiplicative definition?
I feel there must be something of value here. The definition in $(1)$ is very clever. Namely, if you define $x$ as a Dedekind-like cut with left members $x^L$ and $x^R$, and the same for $y$, then you can derive the cut for the product $xy$ using the fact that $x^L<x<r^R$, and similarly $y^L<y<y^R$, which implies the inequalities $$ (x-x^L)(y-y^L)>0,\qquad (x-x^L)(y^R-y)>0, \dots, $$ which implies $xy$ is between the elements of the complicated looking expression in $(1)$. No case splitting is needed. I just cannot quite figure out how to apply that same clever logic to define multiplication of cuts of the already constructed rational numbers.
from Hot Weekly Questions - Mathematics Stack Exchange
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